Suppose each player is randomly and secretly assigned a number between $1$ and $6$ and then a communal dice is rolled four times. After each roll, you can place a bet or fold. After the fourth bet is placed, the person who has the most die facing their number wins.
As each game progresses, the game can move in your favour or against you (e.g., 1/1, 1/2, 2/3, 3/4). If you bet too high at the beginning, the progression of the game could move against you. If you bet too low, it would be difficult to increase the pot at the end because the other players could fold without losing anything. There are infinitely many games available and a finite buy-in amount but an infinite amount of money to replenish the buy-in.
How do you calculate the optimum betting amount?
This relates to one aspect of Texas Hold 'Em poker betting odds, in abstract terms. The other aspects of the betting odds can be ignored.
Any guidelines will be appreciated. I think the question could be simplified to:
Does an initial bet of $10\%$ of capital on $10\%$ odds with a $20\times$ return approach $0$ or $\infty$? What are the limits in this case?